55
4
Physical and Thermal Properties of Cereal Grains
4.1
Introduction
This chapter describes the basic properties of cereal grains that are required for simulating the heat and mass transfer phenomena during drying and storage. Physical dimensions and 1000 grain weight are used to describe the physical characteristics of grains and
their influence on drying. When the drying is simulated by a model using the diffusion
equations to describe internal moisture movement, the preceding parameters are
essential in order to select geometry and specific size. Bulk density, specific heat, latent
heat of vaporization, shrinkage and heat transfer coefficient of a grain bed are essential in
any simulation of the heat and mass transfer during drying. In addition to these properties, a knowledge of thermal conductivity is essential for simulating heat and moisture
movement during storage. The coefficient of friction of cereal grains in themselves
and that on various surfaces are essential for rational design of storage structure.
4.2
Structure of Cereal Grains
A knowledge of structure of cereal grains is useful for interpreting the drying rate in terms
of biological structures. The anatomical structure of all cereal grains is basically similar, but
can differ from one another in details. Grains like wheat, maize, rye and sorghum consist of
pericarp and seed. The seed is composed of testa, embryo and endosperm. Grains like rice,
barley and oats are covered with an additional coat of palea and lemma which constitute
the husk. The anatomical structure of wheat, rice and corn is shown in Figure 4.1.
During drying, moisture flows from the interior to the surface of a kernel, and the flow
of moisture from various parts of the kernel depends on the internal characteristics of the
endosperm, the permeability of the aleurone layer and testa and pericarp and the extent
of damage to these layers.
4.3
Physical Dimensions
Physical dimensions of a cereal grain are of vital importance in the design of cleaning and
grading equipment. Furthermore, the dimensions of a grain have an important influence
on its drying characteristics. Grain breeders need a guide in their work for developing
new varieties of desirable size and shape.
Drying and Storage of Cereal Grains, Second Edition. B. K. Bala.
© 2017 John Wiley & Sons, Ltd. Published 2017 by John Wiley & Sons, Ltd.
56
Drying and Storage of Cereal Grains
(a)
(b)
Brush
Outer pericarp
Aleurone layer
Starchy
endosperm
Nucellar
projection
Pigment strand
Vascular bundle
Pericarp
increase
region
Endosperm
cavity
Scutellum
Coleoptile
Plumule
Epiblast
Primary root
Coleorhiza
Seed coat
Attachment
region
Apex or beard
Hull
Epicarp
Mesocarp
Cross layer
Testa
aleurone layer
Testa
Cross layer
Mesocarp
Epicarp
Starchy
endosperm
Nonflowering
glumes
(c)
Hull
Epidermis
Mesocarp
Cross cells
Tube cells
Seed coat (test)
Aleurone layer
Horny endosperm
Floury endosperm
Cells filled with starch
granules in protein matrix
Walls of cells
Scutellum
Plumule or rudimentary
shoot and leaves
Radicle or primary root
Tip cap.
Figure 4.1 Anatomical structure of (a) wheat, (b) rice and (c) corn.
The length and width of a cereal grain are usually obtained by direct measurement
through an accurate microscope, and the thickness is measured by a micrometer.
4.4 1000 Grain Weight
The 1000 grain weight is also of importance in the design of cleaning and grading equipment and has application in the determination of effective diameter of the grain.
The 1000 grain weight is usually determined by multiplying the weight of 100 randomly selected grains by 10, and the weight of 100 grains is measured by an electronic
precision balance.
Physical and Thermal Properties of Cereal Grains
Example 4.1
The 1000 kernel weight of faba beans with a moisture content of 8.5% (d.b.) is 405 g. Develop
an expression to compute 1000 kernel weight at any moisture content on a dry basis.
Solution
We can write
W1000 = Wd 1 + Md
The preceding equation can be rewritten as
Wd =
W1000
1 + Md
Substituting W1000 = 405 g and Md = 0.085, we have
405
= 373 27g
1 + 0 085
Hence, the required expression is
Wd =
W1000 = 373 27 1 + Md
4.5
Bulk Density
Bulk density is one measure of quality. The bulk density of a cereal grain is usually determined by measuring the weight of a grain sample of known volume. The grain sample is
placed in a cylindrical container of known volume, and the uniform density in the cylinder is obtained by gently tapping the cylinder vertically down onto a table several times
in the same manner. The excess on the top of the cylinder is removed by sliding a string
along the top edge of the cylinder. After the excess has been completely removed, the
weight of the grain sample is measured by an analytical balance. Dry weight of the grain
is determined from the weight of the grain and the moisture content.
4.6
Shrinkage
Shrinkage of an agricultural product during drying is an observable phenomenon, and it
may have significant effect on drying rate and temperature distribution, especially during
deep-bed drying of agricultural crops.
Although several researchers (Boyce, 1966; Nellist, 1974; Spencer, 1972) have reported
that shrinkage is linearly dependent on moisture reduction, Bala (1983) observed that
when a cereal grain such as malt is dried from very high moisture content to very low
moisture content, the shrinkage is not a linear function of moisture reduction, but
the rate of shrinkage decreases with the increase in moisture reduction. Bala (1983) proposed the following hypothesis to predict the shrinkage of a cereal grain on the basis of
experimental data of malt:
The rate of change of shrinkage of grain bed with respect to the reduction in moisture content from initial moisture content is proportional to the difference
between the maximum possible shrinkage and the actual shrinkage.
57
58
Drying and Storage of Cereal Grains
Mathematically,
dy
∞ y0 − y
41
dx
Equation 4.1 can be written as
dy
= ks y0 − y
42
dx
This equation requires the determination of the shrinkage coefficient and the maximum possible shrinkage from experimental data.
The shrinkage of a grain bed is usually determined by simultaneously monitoring the
changes in weight and depth during drying in a bin at constant temperature and mass
flow rate of air (Bala, 1983).
Bala (1983) found reasonable agreement between theory and experiments for malt and
developed the following expression:
S = 1 591 1 −exp − 0 0966 Mw0 − Mw
S E = 0 6871
43
When the equation was used in the simulation models of deep-bed drying of malt, the
agreement between the predicted and experimental temperatures is very good. However,
if the shrinkage is neglected in the models, this caused the models to predict a rapid
passage of the heating front.
Physically, the non-linearity of Equation 4.3 can be interpreted as follows: The shrinkage of a grain bed at any instant during drying is the cumulative effect of the free shrinkage of the cells due to the loss of moisture and elastic shrinkage, if any, due to constraints
on the free shrinkage exerted by the adjacent cells of the grains in the bed. The rate of
shrinkage of the cells in the grains decreases as the moisture content approaches a low
value. This explains why the rate of shrinkage in the grain bed gradually decreases to
almost zero at very low moisture content.
Example 4.2
Malt with a moisture content of 45% (w.b.) was loaded to a depth of 1.0 m in static bed
drier, and it was dried to a moisture content of 3% (w.b.). Determine the final depth of the
grain bed.
Solution
Here, Mw0 = 45% and Mw = 3%
Using Equation 4.3, we have
S = 15 91 1 − exp −0 0966 45−3
= 15 63
The final depth of the grain bed is
1−1 ×
15 63
100
= 0 84 m
4.7 Friction
The coefficient of friction of cereal grains in themselves and that on various surfaces
are essential for rational design of grain bins, silos and other storage structures. These
properties are also important in the design of handling and processing equipment.
Physical and Thermal Properties of Cereal Grains
θi
Figure 4.2 Angle of repose of grains.
4.7.1 Angle of Internal Friction and Angle of Repose
Coefficient of friction between granular materials is the tangent of the angle of internal
friction for that material. The angle of repose is the angle which the side of the piled
materials makes with the horizontal (Figure 4.2). For any material the angle of repose
varies with moisture content and amount of foreign materials present and increases with
the increase in either. Engineers generally assume that the angle of internal friction is
approximately the same as the angle of repose. But some researchers have reported that
there is a difference between the two.
There are two types of angle of repose: (i) static angle of repose and (ii) dynamic angle
of repose. The static angle of repose is the angle of friction taken up by a granular solid
about to slide by itself, whereas the dynamic angle of repose is the angle of repose when
the bulk of material is in motion such as movement of solids from bins.
Static angle of repose of cereal grains can be determined by using a wooden frame full
of grains mounted on a tilting table. The top of the table is tilted until the grain begins to
move along an inclined surface of the grains.
Dynamic angle of repose can be determined by using a specially constructed box which
contains grains. Then the front panel is quickly removed, which allows the grains to flow
to their natural slope.
Angle of internal friction is measured by means of special shearing box in which bulk
material particles slide on each other along a plane. The details of test procedure have
been described by Sitkei (1986).
4.7.2 Coefficient of Friction
When one body is sliding on another, the tangential force resisting the motion is directly
proportional to the normal force between the surfaces in contact. Thus, if F is the friction
force and N is the normal force, as shown in Figure 4.3, then,
F =μ N
44
The coefficient of friction is approximately independent of the areas in contact, the
sliding velocity and the intensity of pressure.
There are two types of friction: (i) static friction and (ii) dynamic friction. A static
friction is encountered at the start of motion, whereas dynamic friction is present during
motion.
59
60
Drying and Storage of Cereal Grains
F = μ′N
θ′
F
F
R
θ
N
N
Figure 4.3 Forces and angle of friction.
W
Sin
θr
θr
θr
θr
W Cos θr
W
Figure 4.4 Device for measuring the angle of static friction.
Let W be the weight of the frame containing grains and ϴr the tilting angle. The resulting force F is given by
F = W sin θr − μ W cos θr
45
At the point of sliding,
0 = W sin θr − μ W cos θr
μ = tan θ
46
Equation 4.6 shows that static friction is the tangent of angle of static friction.
To determine the coefficient of static friction, the material to be tested is fastened to a
tilting table and a small wooden frame filled with grains is placed on the table (Figure 4.4).
The frame is raised slightly so that it does not touch the material. The table is slowly
tilted until the friction force between the grains and the material is overcome by gravity
and downward movement begins. The coefficient of friction is calculated from the
slope angle.
Coefficient of static and dynamic friction can be determined by placing the grains in a
small box in contact with positively driven surface and employing force transducers
(Stewart et al., 1969).
Physical and Thermal Properties of Cereal Grains
4.8
Specific Heat
The specific heat at constant pressure is normally used for studying the heat transfer problems during drying and storage of agricultural crops. The pressure dependence of specific heat is very little for both solids and liquids until extremely high pressure is
encountered. Specific heat is also a function of temperature. However, it has been established that at ordinary temperatures and over temperature intervals which are not too
great, specific heat may be considered as a constant physical property.
The need for data on specific heat of food materials has been recognized as early as
1892. Siebel (1892) proposed that specific heat of food materials can be expressed as
equal to the sum of the specific heat of the solid matter and that of water associated with
the dry solid matter. Siebel proposed the following equations for food materials such as
eggs, meat, fruits and vegetables:
For values above freezing
Cpg = 0 837 + 0 03349Mw
47
For values below freezing
Cpg = 0 837 + 0 01256Mw
48
Thermos flask calorimeters and the method of mixtures are now widely used for the
measurement of specific heats of cereal grains. Whatever the type of calorimeter used
and whether the calorimeter fluid is cooled or heated, the use of grains at room temperature avoids the heat gain or loss during the transfer of the grains into the calorimeter
and the only correction needed is for the final temperature in the calorimeter.
There exists a general agreement among the researchers on the observations of specific
heats of common agricultural crops that the specific heat of wet grain increases linearly
with moisture content. The regression equations of some common agricultural crops are
presented in Table 4.1.
The specific heat of a grain is usually determined by the method of mixtures using distilled water as a calorimeter fluid. The technique consists in determining the temperature
change of water contained in the calorimeter when a known quantity of grain is added to
it at a known different temperature. The specific heat is calculated by solving the following heat balance equation:
Heat loss by grain = heat gained by water and calorimeter
Or
Cpg Wg Tg − Tf = Wc Cpc Tf − Ti + Ww Cpl Tf −Ti
Cpg Wg Tg − Tf = Cpl WE + Ww Tf −Ti
49
An ordinary thermos flask can be used as a calorimeter, and fiber glass insulation
should be added between the vacuum bottle and the outer metal walls of the container.
The specific heat can be determined by dropping the grain directly into the calorimeter.
The water equivalent of the thermos flask calorimeter can be determined by using
materials of known specific heat such as lead shot and distilled water instead of grain,
and then for the material of known specific heat, WE becomes the only unknown in
Equation 4.9.
61
62
Drying and Storage of Cereal Grains
Table 4.1 Specific heat of some common agricultural crops.
Crops
Wheat
Specific heat/regression
equation for specific
heat, kJ/kg K
Author(s)
Remarks
1.594
Babbit (1945)
Determined indirectly
Cpg = 1.184 + 0.03031Mw
Pfalzner (1951)
Sample A
Cpg = 1.260 + 0.03068Mw
Sample B
Cpg = 1.205 + 0.03466Mw
Sample C
Wheat (Soft White)
Cpg = 1.398 + 0.04080Mw
Kazarian and
Hall (1965)
—
Wheat (Hard Red
Spring)
Cpg = 1.096 + 0.04080Md
Muir and
Viravanichai
(1972)
—
Rough rice
Cpg = 1.109 + 0.04479Mw
Haswell (1954)
—
Cpg = 0.921 + 0.05447Mw
Wratten et al.
(1969)
—
Rough rice (short grain) Cpg = 1.269 + 0.03487Mw
Morita and Singh —
(1979)
Rough rice (medium)
Cpg = 1.136 + 0.01758Mw
Vemuganti and
Pfost (1980)
—
Corn (yellow dent)
Cpg = 1.523 + 0.03562Mw
Kazarian and
Hall (1965)
Moisture content
0.91–30.2%
Maize
1.835
Matouk (1976)
Specific heat of dry
matter in the
temperature range
0–15 C
Corn dent
Cpg = 0.77 + 0.00502Mw
Vemuganti and
Pfost (1980)
—
Soybean
Cpg = 1.64 + 0.019Md
Alam and Shove —
(1973)
Barley
Cpg = 0.878 + 0.03475Mw
Vemuganti and
Pfost (1980)
—
Cpg = 1.445 + 0.04885Md
Boyce (1966)
Moisture content
7.70–34.52%
Malt
Cpg = 1.651 + 0.04116Mw
Bala (1983)
Rough rice
Cpg = 1.620 + 0.03114Mw
Bala et al. (1987) Moisture content
9.76–30.44%
Example 4.3
Twenty-five grams of rough rice (moisture content of 13.5% w.b.) at a temperature of
22 C were dropped into a calorimeter containing 46.1 g of ice-cooled water at a temperature of 4.66 C. The final temperature of the mixture is 7.84 C. The water equivalent of
the calorimeter is 17.76 g. Determine the specific heat of rough rice at a moisture content
of 13.5% on a wet basis.
Physical and Thermal Properties of Cereal Grains
Solution
Equation 4.9 can be written as
Cpg =
Cpl WE + Ww Tf −Ti
WE Tg −Tf
We have here Ti = 4.66 C, Tg = 22 C, Tf = 7.84 C, WE = 17.76 g.
Ww = 46.1 g, Wg = 25 g and Cpl = 4.186 kJ/kg K
Substituting these values to the preceding equation gives
Cpg =
4.9
4 186 × 17 76 + 46 1 7 84−4 66
= 2 40kJ kg K
25 × 22−7 84
Thermal Conductivity
Temperature of cereal grains changes during storage as a result of day-to-day variation
in weather conditions, and it is one of the most important factors controlling the rate of
deterioration during storage. To design a storage system rationally, one must predict the
temperature changes during storage. A knowledge of thermal conductivities is essential
for the prediction of grain temperature. Thermal conductivity of grain is also important
in the engineering design of drying, cooling and aeration systems.
There are mainly two methods for determining thermal conductivity of cereal grains.
They are:
1) Steady state method
2) Transient heat flow method.
In steady state method, either a cylindrical apparatus (Babbit, 1945; Bakke and Stiles,
1935) or a spherical apparatus (Oxley, 1944) is used. The transient heat flow method
using a line heating source apparatus developed by Hooper and Lepper (1950) has been
used extensively in the determination of thermal conductivity of cereal grains (Bala et al.,
1987; Chandra and Muir, 1971; Morita and Singh, 1979; Wratten et al., 1969).
The main objections to the steady state method are (i) the long time required to achieve
steady state condition and (ii) the possibility of moisture migration along the temperature gradient during long-test period.
Both of these factors are minimized in the transient heat flow method because of the
reduced-test period. The thermal conductivity of wheat (Chandra and Muir, 1971;
Kazarian and Hall, 1965) and rough rice (Bala et al., 1987; Morita and Singh, 1979;
Wratten et al., 1969) has been reported to increase with the increase in moisture contents and the regression equations developed are shown in Table 4.2.
4.9.1 Theory
The basic equation for the heat flow from a line source is
∂T
∂2 T 1 ∂T
=α
+
∂t
∂r 2 r ∂r
4 10
63
64
Drying and Storage of Cereal Grains
Table 4.2 Thermal conductivity of wheat and rough rice.
Thermal conductivity equation
r
Crops
K = 0.1393 + 0.001196Mw
0.77
Wheat
K = 0.1167 + 0.001130Mw
Authors
Chandra and Muir (1971)
Kazarian and Hall (1965)
K = 0.09999 + 0.01107Mw
0.80
K = 0.0865583 + 0.001327Mw
0.93
Wratten et al. (1969)
K = 0.1193 + 0.001885Mw
0.96
Bala et al. (1987)
Rice
Morita and Singh (1979)
The solution of the change in temperature at a point close to the line heat source
between time t1 and t2 can be written in the following form:
T2 − T1 =
Q
t2
log
4πK e t1
4 11
Thus, the equation for thermal conductivity is
K=
Qloge t2 t1
4π T2 − T1
4 12
There are two sources of error in the use of these equations:
1) In the derivation of this equation, only the first two terms of an infinite series have
been considered. But researchers (Hooper and Lepper, 1950; Kazarian and Hall,
1965) have shown that the error involved due to neglecting the higher-order terms
is negligible.
2) In actual test apparatus, the line heat source used has a finite length and diameter.
Again, the axial heat flow is considered negligible.
To compensate the heat source which in effect replaces a small core of grain, Van der
Held and Van Drumen (1949) have shown that the difference in heat absorption between
the heater and the displaced core can be considered as heat production before the start of
measured time. That is, time t0 is subtracted from each observed time. By differentiation,
Equation 4.12 gives
dT 4πK
=
tc
dt
Q
4 13
at dt/dT = 0, tc = t0 the time correction.
The corrected equation for thermal conductivity thus becomes
K=
Q
t2 −t0
ln
t1 −t0
4π T2 − T1
4 14
Underwood and McTaggart (1960) proposed another method which is simple and
less time-consuming. By using this method, there is no need for evaluating the correction
time. Temperature rise is plotted against time on a semi-log paper. Once the straight-line
Physical and Thermal Properties of Cereal Grains
Figure 4.5 Thermal conductivity
measurement apparatus.
Aluminium
cylinder
Variable
resistance
Heating
wire
DC
supply
Insulation
Ammeter
portion is established, the value obtained from the straight-line portion of the curves is
used in the following equation to calculate the bulk thermal conductivity:
K=
I 2R
θ2
ln
θ1
4π T2 − T1
4 15
Time θ2 and θ1 corresponds to temperature T2 and T1 on the straight-line portion of
the curve on a semi-log paper.
4.9.2 Apparatus and Measurement
The apparatus basically consists of a cylinder made usually of aluminium with a heating
wire stretched between copper leads on the axis of a cylinder. The cylinder is insulated.
A schematic diagram of an apparatus is shown in Figure 4.5.
Heat is supplied by a constant d.c. power through a variable resistance. The current is
measured by an ammeter, and the temperature of the heating wire is measured by a
thermocouple.
To measure thermal conductivity, the cylinder is filled with grain and tapping is performed to obtain uniform density. The temperature at the centre of the cylinder is checked.
Once the temperature is stabilized, current is turned on and the temperature is recorded
at regular intervals. The bulk thermal conductivity is computed using Equation 4.15.
Example 4.4
The temperature–time of a test for determining the bulk thermal conductivity of rough
rice with a moisture content of 13.5% (w.b.) was plotted on a semi-log paper. The temperature difference corresponding to time 130 and 60 s on the linear portion of the semilog plot is 1.8 C. The resistance and the current in the heating wire were 2.458 Ω/m and
1.3 A, respectively. Determine the bulk thermal conductivity.
Solution
Here we have
θ2 = 130 s, θ1 = 60 s, I = 1.3 A, R = 2.458 Ω/m and
T2 − T1 = 1.8 C
Substituting these values in Equation 4.15 yields
K=
2 458 × 1 3 2
ln 130 60 = 0 14W m C
4π × 1 8
65
66
Drying and Storage of Cereal Grains
4.10
Latent Heat of Vaporization of Grain Moisture
The latent heat of vaporization of a grain moisture is defined as the energy required to
vaporize moisture from the product. The energy required to evaporate moisture from
grain especially at low moisture content is higher than that of free water and depends
on the type of crop.
Othmer (1940), starting with the Clapeyron equation, developed the following
equation:
log P =
L
log P + C
L
4 16
where P and P’ are vapour pressures and L and L are molal heats of the two compounds,
respectively, taken at the same temperatures. C is a constant. Equation 4.16 states that, if
the log of the pressure of any substance is plotted against the log of the pressure of any
other substance, a straight line results which will have for its slope the ratio of the molal
latent heats. Plots were made to illustrate the utility of this relation with various materials
in checking and correlating vapour pressure data, and it matched the data.
Gallaher (1951), using the equilibrium moisture content data for wheat (published by
Gay), determined the latent heat of vaporization of wheat and found that when the ratios
of the latent heat of wheat to the latent heat of free water were plotted against moisture
content (d.b.), the resulting curve could be described by the following equation:
Lwheat
= 1 + 23 exp −0 40Md
Lwater
4 17
Bala (1983), using the equilibrium moisture content data for malt from the unpublished data of Pixton and Henderson (1981), developed Othmer plots for malt and also
an equation to describe the ratio of the latent heat of vaporization of malt to the latent
heat of vaporization of free water as a function of moisture content and of the form used
by Gallaher (Equation 4.17). The following regression equation was developed:
Lmalt
= 1 + 0 5904 exp −0 1367Md
Lwater
4 18
Equations 4.17 and 4.18 show that the amount of heat required to vaporize moisture
from the grain increases considerably as the moisture content of the grain decreases.
Table 4.3 shows the equations of latent heats of vaporization of some agricultural crops.
The theoretical basis for the determination of latent heat of a grain is the Othmer plot
based on the Clapeyron equation. The Clapeyron equation is given by Rogers and
Mayhew (1980):
dP
L
=
dTab
V − v Tab
4 19
If v is disregarded in comparison to V and if it is possible to assume a perfect gas, the
following equation may be used for the volume term:
V=
R0 Tab
P
4 20
Physical and Thermal Properties of Cereal Grains
Table 4.3 Latent heat of vaporization of some agricultural crops.
Regression equation of latent heat of
agricultural crops
Crops
Wheat
Shelled corn
Malt
Soybean
Lwheat
= 1 + 23exp − 0 40Md
Lwater
Lcorn
= 1 + 0 8943exp − 0 1232Md
Lwater
Lmalt
= 1 + 0 5953exp − 0 1367Md
Lwater
Lsoybean
= 1 + 0 21624exp − 0 06233Md
Lwater
Authors
Gallahar (1951)
Strohman and Yoeger (1967)
Bala (1983)
Alam and Shove (1973)
From the combination of Equations 4.19 and 4.20, the Clapeyron equation results in
dP
LP
=
4 21
2
dTab R0 Tab
1 dP dTab
=
2
L P R0 Tab
4 22
The same equation may be written for water in grains at the same temperature:
1 dP
dTab
=
2
R0 Tab
L P
4 23
From Equations 4.22 and 4.23
dP P L
=
dP P
L
4 24
It is also apparent that Equation 4.24 can be integrated to give
log P =
L
log P + C
L
4 25
The latent heat of the material can be established from the slope of the log P versus log
P curve.
4.10.1
Determination of Latent Heat of Vaporization of a Grain
The latent heat of vaporization of grain is determined from equilibrium moisture content
data. For each temperature, the saturated vapour pressure is found from the steam tables.
The vapour pressure of a grain at each moisture content is determined by multiplying the
corresponding relative humidity by the saturation vapour pressure for the given temperature. The logarithm of the vapour pressure of grain is plotted against the logarithm of
the vapour pressure of free water. This should give well-defined straight lines. The slope
of the lines, namely, the ratios of the latent heat of grain to the latent heat of free water, is
plotted against moisture content (d.b.). An equation can be developed to describe the
ratio of the latent heat of grain to the latent heat of free water as a function of moisture
content and may be of the form used by Gallaher.
67
Drying and Storage of Cereal Grains
Example 4.5
The equilibrium moisture content–relative humidity data of malt for different temperatures are given in the following table. Draw Othmer plot and determine the ratio of latent
heat of the moisture of malt to latent heat of free water at each moisture content.
Relative humidity, %
Moisture content, % (d.b.)
5 C
25 C
45 C
47.92
38.12
19.18
10.74
7.41
5.59
95.9
95.0
76.1
37.2
17.7
11.2
97.0
95.0
79.3
42.4
21.8
14.9
95.3
94.0
82.0
53.6
30.5
18.6
Solution
The saturation vapour pressures for each of the given temperatures are obtained from
steam tables. By multiplying the equilibrium relative humidity by the saturation vapour
pressure for the given temperature, the values of the grain vapour pressure are obtained.
These vapour pressures are plotted against the vapour pressure of free water on log
paper, and the resulting Othmer plot is shown in Figure E4.5. They give reasonably
5°C
2.0
25°C
45°C
1.6
2.0
1.8
Log10 of vapour pressure from malt, millibar
68
1.6
1.4
1.2
1.0
M.C % (db)
38.12
0.8
19.18
0.6
10.74
0.4
0.2
7.41
5.59
0.0
0.0
0.4
0.8
1.2
Log10 of vapour pressure of water, millibar
Figure E4.5 Othmer plots for equilibrium
moisture content data of malt.
Physical and Thermal Properties of Cereal Grains
well-defined straight lines. The slopes of the lines in Figure E4.5 are the ratio of the
latent heat of grain moisture to the latent heat of free water. The vapour pressure and
the ratios of the latent heats are given in the following table:
Equilibrium vapour pressure, mbar
Moisture, % (d.b.)
5 C
25 C
45 C
Ratio of latent heat of
malt to latent heat of water
47.92
38.12
19.18
10.74
7.41
5.59
Saturation vapour
pressure (mbar)
8.3617
8.2832
6.6353
3.2453
1.5432
0.9765
8.7192
30.7208
30.0874
25.1151
13.4285
6.9042
4.7189
31.6710
91.3498
90.1037
78.6011
51.3782
29.2357
17.8290
95.8550
1.0000
1.0000
1.0410
1.1503
1.2222
1.2500
—
4.11 Heat Transfer Coefficient of Grain Bed
The rate of heat transfer between a solid and a fluid may be computed from the following
relation:
qc = hc AΔT
4 26
Equation 4.26 was originally proposed by Isaac Newton in 1770. Engineers have used
this equation for many years, although it is a definition of hc rather than a phenomenological law of convection. The heat transfer coefficient is actually a complicated function
of the fluid flow, the thermal properties of the fluid medium and the geometry of the
system.
There are mainly two general methods available for the determination of convection
heat transfer coefficients in a packed bed of granular materials:
1) Dimensional analysis correlating existing data
2) Direct measurement of heat transfer coefficient of a grain bed, comparing the temperature curves with Schumann’s exact solution.
Dimensional analysis is mathematically simple and has a wide range of applications.
This method is useless and incomplete without sufficient experimental data and contributes little to our understanding of the transfer process, but it facilitates the interpretation and extends the range of application of experimental data by correlating them in
terms of dimensionless groups.
The second method presupposes that the physical mechanisms can sufficiently be well
understood and described in the form of partial differential equations. Mathematical
solution of the set of partial differential equation is quite complicated. The solution
in the dimensionless form is compared with the experimental data to compute convective heat transfer coefficient. Exact solutions are more important because the assumption
made in this course of analysis can be specified accurately and the validity can be checked
by experiment.
69
70
Drying and Storage of Cereal Grains
4.11.1 Dimensional Analysis
The correlating equation developed by Yoshida et al. (1962) is selected. This equation
was developed by correlation of the data of Wilke and Hougen; Gamson, Thodos and
Hougen; and Wakao, Oshaima and Yagi. Thus, the equation has been supported by sufficient experimental data.
For
50 < Re < 1000
4 27
jh = 0 61Re − 0 41
Again
jh =
hcs Cpa μ
Cpa G K
2 3
4 28
Equating Equations 4.27 and 4.28
hcs = 0 61Cpa G
− 0 41
G
av φμ
Cpa μ
K
−2 3
4 29
where av = 6 1 −ε desp
Assuming ϕ = sp and Cpa/K = 0.735, Equation 4.29 can be reduced to the form
hcv = av hcs = 9 3679Cpa
1−ε
de
1 41 0 41
μ
φ
× G − 0 59
4 30
Example 4.6
The effective diameter of a barley grain is 4.57 mm. The void fraction and shape factor
are 0.51 and 0.9, respectively. Develop an equation of volumetric heat transfer coefficient
for barley bed as a function of mass flow rate of air.
Solution
We have here de = 0.00457 m, ε = 0.51 and ϕ = 0.9. Substituting these values and the
values of Cpa and μ at 77 C in Equation 4.30, we have
hcv = 9 3679 × 1 009
1 −0 51
0 00457
1 41
×
2 075 × 10 −5
0 90
0 41
× G0 59
= 92 03G0 59 kJ m3 s K
4.11.2 Comparison of Theory and Experiment
Schumann (1929) developed a set of partial differential equations for fluid and solid temperature distribution in packed bed of crushed material. The heat transfer coefficient can
be determined by comparing the experimentally determined results with these analytical
solutions. This technique of using Schumann’s curves for determination of heat transfer
coefficient was developed by Furnas (1930), and this technique has also been used by
Physical and Thermal Properties of Cereal Grains
several researchers (Alanis et al., 1977; Bala, 1983; LÖf and Hawley, 1948; Wang
et al., 1978).
Barker (1965) made an extensive survey on heat transfer in packed beds and found, by
plotting the Colburn jh factor against the Reynolds number, that whatever the type of
packing, there is a general agreement among most of the investigators. The agreement
is generally within a factor of about 2 over Reynolds numbers ranging from 10 to 100,000
and especially in the most common range from 200 to 4000.
4.11.2.1
Theory
The formulation of Schumann’s equations by a comparatively simpler method is given in
the following text. The equations are derived for isomoisture heating of grains.
Assumptions are:
1) The air flow is one-dimensional.
2) There is no heat loss perpendicular to the direction of air flow.
3) Direct transfer of heat between particles is negligible.
4) No shrinkage of bed occurs.
5) Thermal properties are constant.
6) Contribution of (δTa/δt) is negligible.
Consider an elemental layer of grain of thickness dz and unit cross section. The data for
zero heat is at 0 C. Then in unit time the heat flowing into the element (z, z + dz) is
GCpa Ta z
and the heat flowing out is
GCpa Ta z + dz
The difference represents the heat transferred convectively to the grain, hcv (Ta − Tg)
dz, and that accumulated in the air volume ρa(δTa/δt)dz. The conservation of heat flow
demands that
∂Ta
dz
4 31
GCpa Ta z + dz − Ta z = −hcv Ta − Tg dz −ρa
∂t
Applying Taylor series expansion and ignoring (δTa/δt) gives
∂Ta
hcv
=−
Ta −Tg
∂z
GCpa
4 32
Consider heat exchange for unit depth over a time increment (t, t + dt). At the beginning of the time step, the grain heat is
ρd Cpg Tg t
and (t + dt)
ρd Cpg Tg t + dt
This change of heat is the result of the convective heat transfer from the air. Therefore,
from the principle of the conservation of heat, applying Taylor series expansion over the
interval dt gives
∂Tg
hcv
=
Ta − Tg
∂t ρd Cpg
4 33
71
72
Drying and Storage of Cereal Grains
If Equations 4.32 and 4.33 are normalized into the standard form of Schumann,
one gets
∂Ta
= Tg −Ta
∂Y
4 34
∂Tg
= Ta −Tg
∂Z
4 35
where
Y=
hcv z
GCpa
4 36
Z=
hcv t
ρd Cpg
4 37
With boundary conditions
Ta 0, Z = Ta0
Tg Y , 0 = Tg 0
This set of partial differential equations can be solved by an exponential approximation
and using central difference values updated by an iteration. Equations 4.34 and 4.35 can
be written in terms of finite difference form as
Ta1 − Tg0 − ΔY
Tg − Tgo
Ta2 − Tg0
e
=
+
Ta0 − Tg0
Ta0 − Tg0
Ta0 − Tg0
1 −e − ΔY
4 38
Tg1 −Tg0 −ΔZ
Ta −Tg0
Tg2 − Tg0
=
+
e
Ta0 − Tg0
Ta0 −Tg0
Ta0 − Tg0
1 −e − ΔZ
4 39
The theoretical non-dimensional air temperatures for different values of Z for values
of Y ranging from 2 to 16 are computed using digital computer, and these are shown in
Figure 4.6.
4.11.3 Determination of Volumetric Heat Transfer Coefficient
Grains are dried to equilibrium moisture content and then cooled to room temperature
in a sealed container. A dummy insulated cylinder is filled with grain to the same depth,
and the air temperature and relative humidity used to dry the original sample are blown
up through the grain. After the air temperature and relative humidity are stabilized, the
dummy cylinder is quickly replaced by the experimental cylinder and the air temperature
at different positions is monitored to determine the experimental non-dimensional air
temperatures.
The method to calculate heat transfer coefficient requires the comparison of the air
temperature at different times and depths from the inlet. The theoretical curves for
non-dimensional air temperatures were plotted against the logarithm of Z for several
values of Y on tracing paper. The experimental non-dimensional air temperatures at a
Physical and Thermal Properties of Cereal Grains
1.0
0.9
0.8
Y=
7
Y=
5
Y=
4
=
Y
=
Y
0.4
6
3
Y
=
=
2
0.5
8
Y=
9
Y=
10
Y=
12
Y=
14
Y=
16
0.6
Y
Ta0 – Tg0
Ta – Tg0
0.7
0.3
0.2
0.1
0.0
1
2
3
4
6
8
10
Z
Figure 4.6 Theoretical non-dimensional air temperatures for different values of Z and for values of Y
ranging from 2 to 16.
particular position are then plotted against the logarithm of time on a separate paper, and
the tracing paper containing the theoretical curves was placed on it. The theoretical
curves were shifted along the time axis of the experimental curves until the experimental
curve lined up with one of the theoretical curves. The Y value of the theoretical curve that
lined up with the experimental curve was used to calculate heat transfer coefficient from
Equation 4.36. Table 4.4 shows equations of transfer coefficient for the grain bed of some
agricultural crops.
Table 4.4 Heat transfer coefficients of grain bed of some agricultural crops.
Crops
Equation of heat transfer
coefficient
Barley
hcv = 89 83 × G − 0 59
Maize
Malt
hcv = 856 8
G Ta + 273
Pat
hcv = 372 6
G × Tab −0 5217
Pat
hcv = 82 25 × G − 0 59
hcv = 49 32 × G
Rice
Wheat
− 0 6011
Authors
Remarks if an
Bala (1983)
Dimensional analysis
Boyce (1966)
Thin layer
experiment
Matouk (1976)
Thin layer
experiment
Bala (1983)
Dimensional analysis
−0 6906
Direct experiment
hcv = 86 9 × G
− 1 30
Wang et al. (1978)
Direct experiment
hcv = 2 24 × G
− 0 494
Henderson and Pabis (1962)
Treybal’s data
73
Drying and Storage of Cereal Grains
Example 4.7
Figure E4.7 shows experimental non-dimensional temperatures of a cereal grain lined up
with non-dimensional theoretical temperatures for three different positions of a grain
bed from an experiment conducted to determine heat transfer coefficient. The mass flow
rate of air in the grain bed is 0.3525 kg/m2 s. Determine the mean value of the heat transfer coefficient.
Solution
Equation 4.36 can be expressed as
YGCpa
z
We have G = 0.3525 kg/m2 s and Cpa = 1.0048 kJ/kg K
hcv =
1) For position z = 0.06, Y = 4.5
hcv =
4 5 × 0 3525 × 1 0048
= 26 56
0 06
2) For position z = 0.12, Y = 8
hcv =
8 × 0 3525 × 1 0048
= 23 61
0 12
3) For position z = 0.18, Y = 12
hcv =
12 × 0 3525 × 1 0048
= 23 61
0 18
Hence the mean volumetric heat coefficient is hcv = 24.59 kJ/s K.
Run-7
1.0
0.9
0.8
14
Y=
16
Y=
8
Y=
9
Y=
10
=
Y
4
3
=
Y
0.3
Y=
Y=
Y
0.4
7
6
2
0.5
Y=
Y=
Y=
5
12
0.6
=
Ta – Tg0
0.7
Ta0 – Tg0
74
0.2
18 cm
12 cm
6 cm
0.1
0.0
1
2
3
4
6
8 10
20
Z
Figure E4.7 Computed and observed temperature history of air.
30